question_answer

The equation of tangents to the hyperbola \[3{{x}^{2}}-2{{y}^{2}}=6,\] which is perpendicular to the line x ? 3y = 3, are

A) \[y=-3x\pm \sqrt{15}\]

B) \[y=3x\pm \sqrt{6}\]

C) \[y=-3x\pm \sqrt{6}\]

D) \[y=2x\pm \sqrt{15}\]

Correct Answer: A

Solution :

Given, equation of hyperbola is                 \[=\left( a+b-c \right).\left[ a\times b-a\times c-b\times b+b\times c \right]\] \[\text{=a}\text{.a }\!\!\times\!\!\text{ b-a}\text{.a }\!\!\times\!\!\text{ c+a}\text{.b }\!\!\times\!\!\text{ c+b}\text{.a }\!\!\times\!\!\text{ b-b}\text{.a}\]            \[\times c+b.b\times c-c.a\times b+c.a\times c-c.b\times c\] On comparing with \[=a.b\times c-b.a\times c-c.a\times b\] we get                           \[=\left[ abc \right]-\left[ bac \right]-\left[ cab \right]\] Given, equation of line is x ? 3y = 3. \[=\left[ abc \right]+\left[ abc \right]-\left[ abc \right]\]  Slope of given line \[=\left[ abc \right]=a.b\times c\] Slope of line perpendicular to given line, m = - 3 Now, the equation of tangents are                  \[A=6{{x}^{2}}.\]                    \[\frac{dA}{dx}=12x\]                    \[\text{ }\!\!\Delta\!\!\text{ x = m }\!\!%\!\!\text{  of x}\]